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It will be seen from the above, that in general a complete series of tones may be produced, corresponding to the complete series of partial tones, 1, 2, 3, 4, &c, up to the generators. It will be noticed also, that the same tones may occur with compound tones, as differential tones of their upper partials.
Though combination tones are generally subjective phenomena, yet on some instruments, as for example, the Double Syren and the Harmonium, they are objective, or at any rate partly so. As a proof of this fact, it is found that differential tones on these instruments, may be strengthened by resonance. Thus, sound loudly G1 and D- on a harmonium, and tune a resonator to the differential G. By alternately applying the resonator to, and withdrawing it from the ear, while the generating notes are being sounded, it is easy to appreciate the alternate reinforcement and falling off of the G.
It was formerly thought, that differential tones were formed by the coalescence of beats (see next Chap.), a supposition which was supported by the fact, that the number of beats generated by two tones in a second, is identical with the vibration number of the differential tone they generate. That this is not the cause of Differential Tones, will be seen from the following considerations: 1st. Under favourable circumstances, the rattle of the beats
and the differential tone may be heard simultaneously. 2nd. Beats are audible, when the generating tones are very faint, in fact, they may be heard even when the generating tones are inaudible. Differentials, on the other hand, invariably require tolerably loud generators. 3rd. This supposition offers no explanation of the origin of the analogous phenomenon of Summation Tones. Finally, Helmholtz has offered a theory of the origin of Differential Tones, which satisfactorily explains all the phenomena of both Differential and Summation tones. This theory is difficult to explain, without such recourse to mathematics, as would be unsuit­able to a work like this. "We must be content, therefore, to state it in general terms as follows:
When two series of sound-waves simultaneously traverse the same mass of air, it is generally assumed that the resultant motion of the air particles is equal to the algebraic sum of the motions, that the air particles would have had, if the two series had traversed the mass of air, independently of one another. This, however, is only strictly true, when the amplitudes of the sound­waves are very small, that is, when the air particles oscillate only through very small spaces. When the amplitudes of the waves are